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This book presents a new research topic in statistics ? vertical
density representation (VDR). The theory of VDR has been found to be
useful for developing new ideas and methodologies in statistics and
management science. The first paper related to VDR appeared in
1991. Several others have since been published and work is continuing
on the topic. The purpose of this book is to survey the results
presented in those papers and provide some new, unpublished results.

VDR may be regarded as a special kind of transformation. By assuming
that a variate is uniformly distributed on the contours of a given
function in real n-dimensional space, and considering the
density of the ordinate of the given function, the density of the
original variate can be represented. The book discusses basic results
and extensions. In particular, the uniform assumption on contours is
relaxed to the general case. Applications are presented in Monte Carlo
simulation, chaos-based uniform random number generation, and what may be
called behavioral estimation. In addition, the authors include a new
result in analyzing correlation into two separate components, which
provides flexibility in modeling correlated phenomena, such as when
combining expert estimates.